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9 THE THEOREM OF FROBENIUS 311

As in the proof of (1 0.9.4) we see that there is a unique solution $-+v(£₉ z, x₀ ₉y_Q)
of that equation, defined for |£| < 2 and such that v(0, z, ;c₀ , y₀) = 0, pro-
vided \\x₀-a\\ <a/8, ||z|| < inf (a/4, £/2M), \\y₀ - b\\ < /?. Furthermore,
(10.7.3) shows that v is continuously differentiate for these values of
£, z, x₀ , y_Q provided a and jS have been taken such that the derivative of
U is bounded in S₀xT₀. Then (10.9.4) shows that wfejc₀,^₀) =
y₀ + v(l , x - x₀ , x₀ , y₀) is the unique solution of (1 0.9.1 ), defined
in S: || x - a\\ ^ a/8, taking the value y₀ for x = X_Q , hence
(x, *o , ^o) -* u(x, x₀ , j₀) is continuously differentiate in S x S x T₀ . If E
and F are finite dimensional, the proof that u is p times continuously
differentiate (resp. indefinitely differentiable, resp. analytic) when U has
the corresponding property, is done in the same way, using (10.7.4) (resp.
(10.7.5)) instead of (10.7.3). Finally, the last statement of the theorem is
proved by the same argument as part (c) of (10.8.1).

When E = K", the Frobenius condition (10.9.4.1) of complete integrability
is equivalent, for the system (10.9.3), to the relations

d d

(10.9.6) -/,(*i, • • • , *„, y) + ft(x_l9...₉x_n9 y) •//*!, ...,*„, y)

(where it must be remembered that — /i(x_l9 . . ., x_tt9 y) is an element of

\ F) (a matrix if F is finite dimensional), and//jc_1? . . . , x_n , y) an element
ofF).



	9 THE THEOREM OF FROBENIUS 311 As in the proof of (1 0.9.4) we see that there is a unique solution $-+v(£₉ z, x₀ ₉y_Q) of that equation, defined for \|£\| < 2 and such that v(0, z, ;c₀ , y₀) = 0, pro- vided \\x₀-a\\ <a/8, \|\|z\|\| < inf (a/4, £/2M), \\y₀ - b\\ < /?. Furthermore, (10.7.3) shows that v is continuously differentiate for these values of £, z, x₀ , y_Q provided a and jS have been taken such that the derivative of U is bounded in S₀xT₀. Then (10.9.4) shows that wfejc₀,^₀) = y₀ + v(l , x - x₀ , x₀ , y₀) is the unique solution of (1 0.9.1 ), defined in S: \|\| x - a\\ ^ a/8, taking the value y₀ for x = X_Q , hence (x, o , ^o) - u(x, x₀ , j₀) is continuously differentiate in S x S x T₀ . If E and F are finite dimensional, the proof that u is p times continuously differentiate (resp. indefinitely differentiable, resp. analytic) when U has the corresponding property, is done in the same way, using (10.7.4) (resp. (10.7.5)) instead of (10.7.3). Finally, the last statement of the theorem is proved by the same argument as part (c) of (10.8.1). When E = K", the Frobenius condition (10.9.4.1) of complete integrability is equivalent, for the system (10.9.3), to the relations d d (10.9.6) -/,(i, • • • , „, y) + ft(x_l9...₉x_n9 y) •//!, ...,„, y)

	(where it must be remembered that — /i(x_l9 . . ., x_tt9 y) is an element of

	\ F) (a matrix if F is finite dimensional), and//jc_1? . . . , x_n , y) an element ofF).